Question

$\sin (\pi +\theta )\sin (\pi -\theta ){\text{cosec}}^2\theta =$

• A. $1$
• B. $-1$
• C. $\sin \theta$
• D. $-\sin \theta$

Answer 1

The correct option is B

$-1$

Explanation for the correct answer:

Simplifying the given expression using trigonometric identities:

Given $\sin (\pi +\theta )\sin (\pi -\theta ){\text{cosec}}^2\theta =$

Applying the trigonometric identities to simplify

$$\begin{gathered} \sin (\pi +\theta )\sin (\pi -\theta ){\text{cosec}}^2\theta \\ =(-\sin \theta )(\sin \theta ){\text{cosec}}^2\theta \left[\because \sin (\pi +\theta )=-\sin \theta \text{and}\sin (\pi -\theta )=\sin \theta \right] \\ =-{\sin }^2\theta {\text{cosec}}^2\theta \\ =-{\sin }^2\theta \left(\frac{1}{{\sin }^2\theta }\right)\left[\because \text{cosec}\left(x\right)=\frac{1}{\sin x}\right] \\ =-1 \end{gathered}$$

Therefore, option (B) is the correct answer.